2020 # Energy convexity of intrinsic bi-harmonic maps and applications I: Spherical target

**Abstract:** In this paper, we show an energy convexity and thus uniqueness for weakly intrinsic bi-harmonic maps from the unit 4-ball {B_{1}\subset\mathbb{R}^{4}} into the sphere {\mathbb{S}^{n}}. In particular, this yields a version of uniqueness of weakly harmonic maps on the unit 4-ball which is new. We also show a version of energy convexity along the intrinsic bi-harmonic map heat flow into {\mathbb{S}^{n}}, which in particular yields the long-time existence of the intrinsic bi-harmonic map heat flow, a result that w…

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“…A similar L p -theory for general even order linear elliptic systems proposed by de Longueville and Gastel [9] was also established by [19] in critical dimensions. For applications to biharmonic maps, see Laurain-Lin [27] for energy convexity and Laurain-Rivière [28] and Wang-Zheng [49] for energy quantization. We also point out that the theory of biharmonic maps has been successfully applied in Cheng-Zhou's solution of the Rosenberg-Smith conjecture in their recent work [7].…”

confidence: 99%

“…A similar L p -theory for general even order linear elliptic systems proposed by de Longueville and Gastel [9] was also established by [19] in critical dimensions. For applications to biharmonic maps, see Laurain-Lin [27] for energy convexity and Laurain-Rivière [28] and Wang-Zheng [49] for energy quantization. We also point out that the theory of biharmonic maps has been successfully applied in Cheng-Zhou's solution of the Rosenberg-Smith conjecture in their recent work [7].…”

confidence: 99%

“…Higher order geometric variational problems, including the study of (both extrinsic and intrinsic) biharmonic maps and polyharmonic maps, have attracted much attention in the last two decades, see e.g. [10,5,42,44,29,39,41,15,30,14,34,1,18,9] for the extrinsic case and [44,25,37,15,32] for the intrinsic case. The corresponding heat flows have been studied extensively as well, as a tool to prove the existence of biharmonic maps and polyharmonic maps in a given homotopy class, see e.g.…”

confidence: 99%

“…The corresponding heat flows have been studied extensively as well, as a tool to prove the existence of biharmonic maps and polyharmonic maps in a given homotopy class, see e.g. [26,27,13,45,20,21,32]. It should be noted that the extrinsic and intrinsic cases come in two different flavors: the intrinsic variants are considered more geometrically natural because they do not depend on the embedding of the target manifold N ֒→ R K , although they are less natural from the variational point of view due to the lack of coercivity for the intrinsic energies (and thus they are considerably more difficult analytically and less studied); on the other hand, the extrinsic variants are more natural from the analytical point of view but in turn they do depend on the embedding of N ֒→ R K .…”

confidence: 99%